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d A B = ( cos ( α β ) 1 ) 2 + ( sin ( α β ) 0 ) 2 = cos 2 ( α β ) 2 cos ( α β ) + 1 + sin 2 ( α β )

Applying the Pythagorean identity and simplifying we get:

= ( cos 2 ( α β ) + sin 2 ( α β ) ) 2 cos ( α β ) + 1 = 1 2 cos ( α β ) + 1 = 2 2 cos ( α β )

Because the two distances are the same, we set them equal to each other and simplify.

2 2 cos α cos β 2 sin α sin β = 2 2 cos ( α β ) 2 2 cos α cos β 2 sin α sin β = 2 2 cos ( α β )

Finally we subtract 2 from both sides and divide both sides by −2.

cos α cos β + sin α sin β = cos ( α β )   

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

Sum and difference formulas for cosine

These formulas can be used to calculate the cosine of sums and differences of angles.

cos ( α + β ) = cos α cos β sin α sin β
cos ( α β ) = cos α cos β + sin α sin β

Given two angles, find the cosine of the difference between the angles.

  1. Write the difference formula for cosine.
  2. Substitute the values of the given angles into the formula.
  3. Simplify.

Finding the exact value using the formula for the cosine of the difference of two angles

Using the formula for the cosine of the difference of two angles, find the exact value of cos ( 5 π 4 π 6 ) .

Begin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.

cos ( α β ) = cos α cos β + sin α sin β cos ( 5 π 4 π 6 ) = cos ( 5 π 4 ) cos ( π 6 ) + sin ( 5 π 4 ) sin ( π 6 ) = ( 2 2 ) ( 3 2 ) ( 2 2 ) ( 1 2 ) = 6 4 2 4 = 6 2 4

Keep in mind that we can always check the answer using a graphing calculator in radian mode.

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Find the exact value of cos ( π 3 π 4 ) .

2 + 6 4

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Finding the exact value using the formula for the sum of two angles for cosine

Find the exact value of cos ( 75° ) .

As 75° = 45° + 30° , we can evaluate cos ( 75° ) as cos ( 45° + 30° ) .

cos ( α + β ) = cos α cos β sin α sin β cos ( 45° + 30° ) = cos ( 45° ) cos ( 30° ) sin ( 45° ) sin ( 30° ) = 2 2 ( 3 2 ) 2 2 ( 1 2 ) = 6 4 2 4 = 6 2 4

Keep in mind that we can always check the answer using a graphing calculator in degree mode.

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Find the exact value of cos ( 105° ) .

2 6 4

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Using the sum and difference formulas for sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

Sum and difference formulas for sine

These formulas can be used to calculate the sines of sums and differences of angles.

sin ( α + β ) = sin α cos β + cos α sin β
sin ( α β ) = sin α cos β cos α sin β

Given two angles, find the sine of the difference between the angles.

  1. Write the difference formula for sine.
  2. Substitute the given angles into the formula.
  3. Simplify.

Using sum and difference identities to evaluate the difference of angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

  1. sin ( 45° 30° )
  2. sin ( 135° 120° )
  1. Let’s begin by writing the formula and substitute the given angles.
    sin ( α β ) = sin α cos β cos α sin β sin ( 45° 30° ) = sin ( 45° ) cos ( 30° ) cos ( 45° ) sin ( 30° )

    Next, we need to find the values of the trigonometric expressions.

    sin ( 45° ) = 2 2 ,   cos ( 30° ) = 3 2 ,   cos ( 45° ) = 2 2 ,   sin ( 30° ) = 1 2

    Now we can substitute these values into the equation and simplify.

    sin ( 45° 30° ) = 2 2 ( 3 2 ) 2 2 ( 1 2 ) = 6 2 4
  2. Again, we write the formula and substitute the given angles.
    sin ( α β ) = sin α cos β cos α sin β sin ( 135° 120° ) = sin ( 135° ) cos ( 120° ) cos ( 135° ) sin ( 120° )

    Next, we find the values of the trigonometric expressions.

    sin ( 135° ) = 2 2 , cos ( 120° ) = 1 2 , cos ( 135° ) = 2 2 , sin ( 120° ) = 3 2

    Now we can substitute these values into the equation and simplify.

    sin ( 135° 120° ) = 2 2 ( 1 2 ) ( 2 2 ) ( 3 2 ) = 2 + 6 4 = 6 2 4 sin ( 135° 120° ) = 2 2 ( 1 2 ) ( 2 2 ) ( 3 2 ) = 2 + 6 4 = 6 2 4
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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